3.1.86 \(\int \frac {1}{\sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^3} \, dx\) [86]

Optimal. Leaf size=476 \[ -\frac {b^{3/2} \left (35 a^4+6 a^2 b^2+3 b^4\right ) \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{4 a^{5/2} \left (a^2+b^2\right )^3 d \sqrt {e}}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d \sqrt {e}}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d \sqrt {e}}-\frac {b^2 \sqrt {e \cot (c+d x)}}{2 a \left (a^2+b^2\right ) d e (a+b \cot (c+d x))^2}-\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {e \cot (c+d x)}}{4 a^2 \left (a^2+b^2\right )^2 d e (a+b \cot (c+d x))}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d \sqrt {e}}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d \sqrt {e}} \]

[Out]

-1/4*b^(3/2)*(35*a^4+6*a^2*b^2+3*b^4)*arctan(b^(1/2)*(e*cot(d*x+c))^(1/2)/a^(1/2)/e^(1/2))/a^(5/2)/(a^2+b^2)^3
/d/e^(1/2)+1/2*(a+b)*(a^2-4*a*b+b^2)*arctan(1-2^(1/2)*(e*cot(d*x+c))^(1/2)/e^(1/2))/(a^2+b^2)^3/d*2^(1/2)/e^(1
/2)-1/2*(a+b)*(a^2-4*a*b+b^2)*arctan(1+2^(1/2)*(e*cot(d*x+c))^(1/2)/e^(1/2))/(a^2+b^2)^3/d*2^(1/2)/e^(1/2)+1/4
*(a-b)*(a^2+4*a*b+b^2)*ln(e^(1/2)+cot(d*x+c)*e^(1/2)-2^(1/2)*(e*cot(d*x+c))^(1/2))/(a^2+b^2)^3/d*2^(1/2)/e^(1/
2)-1/4*(a-b)*(a^2+4*a*b+b^2)*ln(e^(1/2)+cot(d*x+c)*e^(1/2)+2^(1/2)*(e*cot(d*x+c))^(1/2))/(a^2+b^2)^3/d*2^(1/2)
/e^(1/2)-1/2*b^2*(e*cot(d*x+c))^(1/2)/a/(a^2+b^2)/d/e/(a+b*cot(d*x+c))^2-1/4*b^2*(11*a^2+3*b^2)*(e*cot(d*x+c))
^(1/2)/a^2/(a^2+b^2)^2/d/e/(a+b*cot(d*x+c))

________________________________________________________________________________________

Rubi [A]
time = 0.82, antiderivative size = 476, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 13, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {3650, 3730, 3734, 3615, 1182, 1176, 631, 210, 1179, 642, 3715, 65, 211} \begin {gather*} \frac {(a+b) \left (a^2-4 a b+b^2\right ) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d \sqrt {e} \left (a^2+b^2\right )^3}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} d \sqrt {e} \left (a^2+b^2\right )^3}-\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {e \cot (c+d x)}}{4 a^2 d e \left (a^2+b^2\right )^2 (a+b \cot (c+d x))}-\frac {b^2 \sqrt {e \cot (c+d x)}}{2 a d e \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d \sqrt {e} \left (a^2+b^2\right )^3}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d \sqrt {e} \left (a^2+b^2\right )^3}-\frac {b^{3/2} \left (35 a^4+6 a^2 b^2+3 b^4\right ) \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{4 a^{5/2} d \sqrt {e} \left (a^2+b^2\right )^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[e*Cot[c + d*x]]*(a + b*Cot[c + d*x])^3),x]

[Out]

-1/4*(b^(3/2)*(35*a^4 + 6*a^2*b^2 + 3*b^4)*ArcTan[(Sqrt[b]*Sqrt[e*Cot[c + d*x]])/(Sqrt[a]*Sqrt[e])])/(a^(5/2)*
(a^2 + b^2)^3*d*Sqrt[e]) + ((a + b)*(a^2 - 4*a*b + b^2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(S
qrt[2]*(a^2 + b^2)^3*d*Sqrt[e]) - ((a + b)*(a^2 - 4*a*b + b^2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[
e]])/(Sqrt[2]*(a^2 + b^2)^3*d*Sqrt[e]) - (b^2*Sqrt[e*Cot[c + d*x]])/(2*a*(a^2 + b^2)*d*e*(a + b*Cot[c + d*x])^
2) - (b^2*(11*a^2 + 3*b^2)*Sqrt[e*Cot[c + d*x]])/(4*a^2*(a^2 + b^2)^2*d*e*(a + b*Cot[c + d*x])) + ((a - b)*(a^
2 + 4*a*b + b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*(a^2 + b^2)^3*
d*Sqrt[e]) - ((a - b)*(a^2 + 4*a*b + b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/
(2*Sqrt[2]*(a^2 + b^2)^3*d*Sqrt[e])

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 1182

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a*c, 2]}, Dist[(d*q + a*e)/(2*a*c),
Int[(q + c*x^2)/(a + c*x^4), x], x] + Dist[(d*q - a*e)/(2*a*c), Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ
[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[(-a)*c]

Rule 3615

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[2/f, Subst[I
nt[(b*c + d*x^2)/(b^2 + x^4), x], x, Sqrt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2,
0] && NeQ[c^2 + d^2, 0]

Rule 3650

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[b^2*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(a^2 + b^2)*(b*c - a*d))), x] + D
ist[1/((m + 1)*(a^2 + b^2)*(b*c - a*d)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[a*(b*c -
a*d)*(m + 1) - b^2*d*(m + n + 2) - b*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b^2*d*(m + n + 2)*Tan[e + f*x]^2, x],
x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && I
ntegerQ[2*m] && LtQ[m, -1] && (LtQ[n, 0] || IntegerQ[m]) &&  !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] &&
NeQ[a, 0])))

Rule 3715

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_) + (C_.)*
tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Dist[A/f, Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, Tan[e + f*x]], x]
/; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A, C]

Rule 3730

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*t
an[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 - a*(b*B - a*C))*(a + b*Ta
n[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2))), x] + Dist[1/((m + 1)*(
b*c - a*d)*(a^2 + b^2)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1)
- b^2*d*(m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d)*(A*b - a*B - b*C)*Tan[e
+ f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C,
n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] &&  !(ILtQ[n, -1] && ( !I
ntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))

Rule 3734

Int[(((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (
f_.)*(x_)]^2))/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[1/(a^2 + b^2), Int[(c + d*Tan[e + f*
x])^n*Simp[b*B + a*(A - C) + (a*B - b*(A - C))*Tan[e + f*x], x], x], x] + Dist[(A*b^2 - a*b*B + a^2*C)/(a^2 +
b^2), Int[(c + d*Tan[e + f*x])^n*((1 + Tan[e + f*x]^2)/(a + b*Tan[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e,
f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] &&  !GtQ[n, 0] &&  !LeQ[n, -
1]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^3} \, dx &=-\frac {b^2 \sqrt {e \cot (c+d x)}}{2 a \left (a^2+b^2\right ) d e (a+b \cot (c+d x))^2}-\frac {\int \frac {-\frac {1}{2} \left (4 a^2+3 b^2\right ) e+2 a b e \cot (c+d x)-\frac {3}{2} b^2 e \cot ^2(c+d x)}{\sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2} \, dx}{2 a \left (a^2+b^2\right ) e}\\ &=-\frac {b^2 \sqrt {e \cot (c+d x)}}{2 a \left (a^2+b^2\right ) d e (a+b \cot (c+d x))^2}-\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {e \cot (c+d x)}}{4 a^2 \left (a^2+b^2\right )^2 d e (a+b \cot (c+d x))}+\frac {\int \frac {\frac {1}{4} \left (8 a^4+3 a^2 b^2+3 b^4\right ) e^2-4 a^3 b e^2 \cot (c+d x)+\frac {1}{4} b^2 \left (11 a^2+3 b^2\right ) e^2 \cot ^2(c+d x)}{\sqrt {e \cot (c+d x)} (a+b \cot (c+d x))} \, dx}{2 a^2 \left (a^2+b^2\right )^2 e^2}\\ &=-\frac {b^2 \sqrt {e \cot (c+d x)}}{2 a \left (a^2+b^2\right ) d e (a+b \cot (c+d x))^2}-\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {e \cot (c+d x)}}{4 a^2 \left (a^2+b^2\right )^2 d e (a+b \cot (c+d x))}+\frac {\left (b^2 \left (35 a^4+6 a^2 b^2+3 b^4\right )\right ) \int \frac {1+\cot ^2(c+d x)}{\sqrt {e \cot (c+d x)} (a+b \cot (c+d x))} \, dx}{8 a^2 \left (a^2+b^2\right )^3}+\frac {\int \frac {2 a^3 \left (a^2-3 b^2\right ) e^2-2 a^2 b \left (3 a^2-b^2\right ) e^2 \cot (c+d x)}{\sqrt {e \cot (c+d x)}} \, dx}{2 a^2 \left (a^2+b^2\right )^3 e^2}\\ &=-\frac {b^2 \sqrt {e \cot (c+d x)}}{2 a \left (a^2+b^2\right ) d e (a+b \cot (c+d x))^2}-\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {e \cot (c+d x)}}{4 a^2 \left (a^2+b^2\right )^2 d e (a+b \cot (c+d x))}+\frac {\left (b^2 \left (35 a^4+6 a^2 b^2+3 b^4\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-e x} (a-b x)} \, dx,x,-\cot (c+d x)\right )}{8 a^2 \left (a^2+b^2\right )^3 d}+\frac {\text {Subst}\left (\int \frac {-2 a^3 \left (a^2-3 b^2\right ) e^3+2 a^2 b \left (3 a^2-b^2\right ) e^2 x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{a^2 \left (a^2+b^2\right )^3 d e^2}\\ &=-\frac {b^2 \sqrt {e \cot (c+d x)}}{2 a \left (a^2+b^2\right ) d e (a+b \cot (c+d x))^2}-\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {e \cot (c+d x)}}{4 a^2 \left (a^2+b^2\right )^2 d e (a+b \cot (c+d x))}-\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {e+x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{\left (a^2+b^2\right )^3 d}-\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {e-x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{\left (a^2+b^2\right )^3 d}-\frac {\left (b^2 \left (35 a^4+6 a^2 b^2+3 b^4\right )\right ) \text {Subst}\left (\int \frac {1}{a+\frac {b x^2}{e}} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{4 a^2 \left (a^2+b^2\right )^3 d e}\\ &=-\frac {b^{3/2} \left (35 a^4+6 a^2 b^2+3 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{4 a^{5/2} \left (a^2+b^2\right )^3 d \sqrt {e}}-\frac {b^2 \sqrt {e \cot (c+d x)}}{2 a \left (a^2+b^2\right ) d e (a+b \cot (c+d x))^2}-\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {e \cot (c+d x)}}{4 a^2 \left (a^2+b^2\right )^2 d e (a+b \cot (c+d x))}-\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{e-\sqrt {2} \sqrt {e} x+x^2} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{2 \left (a^2+b^2\right )^3 d}-\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{e+\sqrt {2} \sqrt {e} x+x^2} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{2 \left (a^2+b^2\right )^3 d}+\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {e}+2 x}{-e-\sqrt {2} \sqrt {e} x-x^2} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d \sqrt {e}}+\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {e}-2 x}{-e+\sqrt {2} \sqrt {e} x-x^2} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d \sqrt {e}}\\ &=-\frac {b^{3/2} \left (35 a^4+6 a^2 b^2+3 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{4 a^{5/2} \left (a^2+b^2\right )^3 d \sqrt {e}}-\frac {b^2 \sqrt {e \cot (c+d x)}}{2 a \left (a^2+b^2\right ) d e (a+b \cot (c+d x))^2}-\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {e \cot (c+d x)}}{4 a^2 \left (a^2+b^2\right )^2 d e (a+b \cot (c+d x))}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d \sqrt {e}}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d \sqrt {e}}-\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d \sqrt {e}}+\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d \sqrt {e}}\\ &=-\frac {b^{3/2} \left (35 a^4+6 a^2 b^2+3 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{4 a^{5/2} \left (a^2+b^2\right )^3 d \sqrt {e}}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d \sqrt {e}}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d \sqrt {e}}-\frac {b^2 \sqrt {e \cot (c+d x)}}{2 a \left (a^2+b^2\right ) d e (a+b \cot (c+d x))^2}-\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {e \cot (c+d x)}}{4 a^2 \left (a^2+b^2\right )^2 d e (a+b \cot (c+d x))}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d \sqrt {e}}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d \sqrt {e}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 6.14, size = 411, normalized size = 0.86 \begin {gather*} -\frac {\sqrt {\cot (c+d x)} \left (\frac {2 b^{3/2} \left (3 a^2-b^2\right ) \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {\cot (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} \left (a^2+b^2\right )^3}+\frac {2 b^2 \sqrt {\cot (c+d x)} \left (\frac {\sqrt {a} \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {\cot (c+d x)}}{\sqrt {a}}\right )}{\sqrt {b} \sqrt {\cot (c+d x)}}+\frac {a}{a+b \cot (c+d x)}\right )}{a \left (a^2+b^2\right )^2}+\frac {2 b^2 \sqrt {\cot (c+d x)} \, _2F_1\left (\frac {1}{2},3;\frac {3}{2};-\frac {b \cot (c+d x)}{a}\right )}{a^3 \left (a^2+b^2\right )}-\frac {2 b \left (3 a^2-b^2\right ) \cot ^{\frac {3}{2}}(c+d x) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\cot ^2(c+d x)\right )}{3 \left (a^2+b^2\right )^3}-\frac {a \left (a^2-3 b^2\right ) \left (4 \left (\sqrt {2} \text {ArcTan}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )-\sqrt {2} \text {ArcTan}\left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )\right )+2 \sqrt {2} \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )-2 \sqrt {2} \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )\right )}{8 \left (a^2+b^2\right )^3}\right )}{d \sqrt {e \cot (c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[e*Cot[c + d*x]]*(a + b*Cot[c + d*x])^3),x]

[Out]

-((Sqrt[Cot[c + d*x]]*((2*b^(3/2)*(3*a^2 - b^2)*ArcTan[(Sqrt[b]*Sqrt[Cot[c + d*x]])/Sqrt[a]])/(Sqrt[a]*(a^2 +
b^2)^3) + (2*b^2*Sqrt[Cot[c + d*x]]*((Sqrt[a]*ArcTan[(Sqrt[b]*Sqrt[Cot[c + d*x]])/Sqrt[a]])/(Sqrt[b]*Sqrt[Cot[
c + d*x]]) + a/(a + b*Cot[c + d*x])))/(a*(a^2 + b^2)^2) + (2*b^2*Sqrt[Cot[c + d*x]]*Hypergeometric2F1[1/2, 3,
3/2, -((b*Cot[c + d*x])/a)])/(a^3*(a^2 + b^2)) - (2*b*(3*a^2 - b^2)*Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/4,
1, 7/4, -Cot[c + d*x]^2])/(3*(a^2 + b^2)^3) - (a*(a^2 - 3*b^2)*(4*(Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x
]]] - Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]) + 2*Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c +
d*x]] - 2*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/(8*(a^2 + b^2)^3)))/(d*Sqrt[e*Cot[c +
d*x]]))

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Maple [A]
time = 0.71, size = 465, normalized size = 0.98

method	result	size
derivativedivides	\(-\frac {2 e^{4} \left (\frac {b^{2} \left (\frac {\frac {b \left (11 a^{4}+14 a^{2} b^{2}+3 b^{4}\right ) \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{8 a^{2}}+\frac {e \left (13 a^{4}+18 a^{2} b^{2}+5 b^{4}\right ) \sqrt {e \cot \left (d x +c \right )}}{8 a}}{\left (e \cot \left (d x +c \right ) b +a e \right )^{2}}+\frac {\left (35 a^{4}+6 a^{2} b^{2}+3 b^{4}\right ) \arctan \left (\frac {b \sqrt {e \cot \left (d x +c \right )}}{\sqrt {a e b}}\right )}{8 a^{2} \sqrt {a e b}}\right )}{e^{4} \left (a^{2}+b^{2}\right )^{3}}+\frac {\frac {\left (a^{3} e -3 a \,b^{2} e \right ) \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{2}}+\frac {\left (-3 a^{2} b +b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}}{e^{4} \left (a^{2}+b^{2}\right )^{3}}\right )}{d}\)	\(465\)
default	\(-\frac {2 e^{4} \left (\frac {b^{2} \left (\frac {\frac {b \left (11 a^{4}+14 a^{2} b^{2}+3 b^{4}\right ) \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{8 a^{2}}+\frac {e \left (13 a^{4}+18 a^{2} b^{2}+5 b^{4}\right ) \sqrt {e \cot \left (d x +c \right )}}{8 a}}{\left (e \cot \left (d x +c \right ) b +a e \right )^{2}}+\frac {\left (35 a^{4}+6 a^{2} b^{2}+3 b^{4}\right ) \arctan \left (\frac {b \sqrt {e \cot \left (d x +c \right )}}{\sqrt {a e b}}\right )}{8 a^{2} \sqrt {a e b}}\right )}{e^{4} \left (a^{2}+b^{2}\right )^{3}}+\frac {\frac {\left (a^{3} e -3 a \,b^{2} e \right ) \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{2}}+\frac {\left (-3 a^{2} b +b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}}{e^{4} \left (a^{2}+b^{2}\right )^{3}}\right )}{d}\)	\(465\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*cot(d*x+c))^(1/2)/(a+b*cot(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

-2/d*e^4*(b^2/e^4/(a^2+b^2)^3*((1/8*b*(11*a^4+14*a^2*b^2+3*b^4)/a^2*(e*cot(d*x+c))^(3/2)+1/8*e*(13*a^4+18*a^2*
b^2+5*b^4)/a*(e*cot(d*x+c))^(1/2))/(e*cot(d*x+c)*b+a*e)^2+1/8*(35*a^4+6*a^2*b^2+3*b^4)/a^2/(a*e*b)^(1/2)*arcta
n(b*(e*cot(d*x+c))^(1/2)/(a*e*b)^(1/2)))+1/e^4/(a^2+b^2)^3*(1/8*(a^3*e-3*a*b^2*e)*(e^2)^(1/4)/e^2*2^(1/2)*(ln(
(e*cot(d*x+c)+(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)*2^(1/2)+(e^2)^(1/2))/(e*cot(d*x+c)-(e^2)^(1/4)*(e*cot(d*x+c))^(
1/2)*2^(1/2)+(e^2)^(1/2)))+2*arctan(2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1)-2*arctan(-2^(1/2)/(e^2)^(1/4)*
(e*cot(d*x+c))^(1/2)+1))+1/8*(-3*a^2*b+b^3)/(e^2)^(1/4)*2^(1/2)*(ln((e*cot(d*x+c)-(e^2)^(1/4)*(e*cot(d*x+c))^(
1/2)*2^(1/2)+(e^2)^(1/2))/(e*cot(d*x+c)+(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)*2^(1/2)+(e^2)^(1/2)))+2*arctan(2^(1/2
)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1)-2*arctan(-2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1))))

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Maxima [A]
time = 0.52, size = 427, normalized size = 0.90 \begin {gather*} -\frac {{\left (\frac {{\left (35 \, a^{4} b^{2} + 6 \, a^{2} b^{4} + 3 \, b^{6}\right )} \arctan \left (\frac {b}{\sqrt {a b} \sqrt {\tan \left (d x + c\right )}}\right )}{{\left (a^{8} + 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} + a^{2} b^{6}\right )} \sqrt {a b}} + \frac {2 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 2 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) - \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {\frac {13 \, a^{3} b^{2} + 5 \, a b^{4}}{\sqrt {\tan \left (d x + c\right )}} + \frac {11 \, a^{2} b^{3} + 3 \, b^{5}}{\tan \left (d x + c\right )^{\frac {3}{2}}}}{a^{8} + 2 \, a^{6} b^{2} + a^{4} b^{4} + \frac {2 \, {\left (a^{7} b + 2 \, a^{5} b^{3} + a^{3} b^{5}\right )}}{\tan \left (d x + c\right )} + \frac {a^{6} b^{2} + 2 \, a^{4} b^{4} + a^{2} b^{6}}{\tan \left (d x + c\right )^{2}}}\right )} e^{\left (-\frac {1}{2}\right )}}{4 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*cot(d*x+c))^(1/2)/(a+b*cot(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/4*((35*a^4*b^2 + 6*a^2*b^4 + 3*b^6)*arctan(b/(sqrt(a*b)*sqrt(tan(d*x + c))))/((a^8 + 3*a^6*b^2 + 3*a^4*b^4
+ a^2*b^6)*sqrt(a*b)) + (2*sqrt(2)*(a^3 - 3*a^2*b - 3*a*b^2 + b^3)*arctan(1/2*sqrt(2)*(sqrt(2) + 2/sqrt(tan(d*
x + c)))) + 2*sqrt(2)*(a^3 - 3*a^2*b - 3*a*b^2 + b^3)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2/sqrt(tan(d*x + c)))) +
sqrt(2)*(a^3 + 3*a^2*b - 3*a*b^2 - b^3)*log(sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1) - sqrt(2)*(a^3 +
3*a^2*b - 3*a*b^2 - b^3)*log(-sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1))/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 +
b^6) + ((13*a^3*b^2 + 5*a*b^4)/sqrt(tan(d*x + c)) + (11*a^2*b^3 + 3*b^5)/tan(d*x + c)^(3/2))/(a^8 + 2*a^6*b^2
+ a^4*b^4 + 2*(a^7*b + 2*a^5*b^3 + a^3*b^5)/tan(d*x + c) + (a^6*b^2 + 2*a^4*b^4 + a^2*b^6)/tan(d*x + c)^2))*e
^(-1/2)/d

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*cot(d*x+c))^(1/2)/(a+b*cot(d*x+c))^3,x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {e \cot {\left (c + d x \right )}} \left (a + b \cot {\left (c + d x \right )}\right )^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*cot(d*x+c))**(1/2)/(a+b*cot(d*x+c))**3,x)

[Out]

Integral(1/(sqrt(e*cot(c + d*x))*(a + b*cot(c + d*x))**3), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*cot(d*x+c))^(1/2)/(a+b*cot(d*x+c))^3,x, algorithm="giac")

[Out]

integrate(1/((b*cot(d*x + c) + a)^3*sqrt(e*cot(d*x + c))), x)

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Mupad [B]
time = 6.79, size = 2500, normalized size = 5.25 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((e*cot(c + d*x))^(1/2)*(a + b*cot(c + d*x))^3),x)

[Out]

atan(((1i/(4*(b^6*d^2*e - a^6*d^2*e - 15*a^2*b^4*d^2*e - a^3*b^3*d^2*e*20i + 15*a^4*b^2*d^2*e + a*b^5*d^2*e*6i
+ a^5*b*d^2*e*6i)))^(1/2)*((1i/(4*(b^6*d^2*e - a^6*d^2*e - 15*a^2*b^4*d^2*e - a^3*b^3*d^2*e*20i + 15*a^4*b^2*
d^2*e + a*b^5*d^2*e*6i + a^5*b*d^2*e*6i)))^(1/2)*((1i/(4*(b^6*d^2*e - a^6*d^2*e - 15*a^2*b^4*d^2*e - a^3*b^3*d
^2*e*20i + 15*a^4*b^2*d^2*e + a*b^5*d^2*e*6i + a^5*b*d^2*e*6i)))^(1/2)*((1i/(4*(b^6*d^2*e - a^6*d^2*e - 15*a^2
*b^4*d^2*e - a^3*b^3*d^2*e*20i + 15*a^4*b^2*d^2*e + a*b^5*d^2*e*6i + a^5*b*d^2*e*6i)))^(1/2)*((192*a^2*b^24*d^
4*e^10 + 1728*a^4*b^22*d^4*e^10 + 8320*a^6*b^20*d^4*e^10 + 27264*a^8*b^18*d^4*e^10 + 62592*a^10*b^16*d^4*e^10
+ 99456*a^12*b^14*d^4*e^10 + 107520*a^14*b^12*d^4*e^10 + 76800*a^16*b^10*d^4*e^10 + 33984*a^18*b^8*d^4*e^10 +
7872*a^20*b^6*d^4*e^10 + 384*a^22*b^4*d^4*e^10 - 128*a^24*b^2*d^4*e^10)/(a^20*d^5 + a^4*b^16*d^5 + 8*a^6*b^14*
d^5 + 28*a^8*b^12*d^5 + 56*a^10*b^10*d^5 + 70*a^12*b^8*d^5 + 56*a^14*b^6*d^5 + 28*a^16*b^4*d^5 + 8*a^18*b^2*d^
5) - ((1i/(4*(b^6*d^2*e - a^6*d^2*e - 15*a^2*b^4*d^2*e - a^3*b^3*d^2*e*20i + 15*a^4*b^2*d^2*e + a*b^5*d^2*e*6i
+ a^5*b*d^2*e*6i)))^(1/2)*(e*cot(c + d*x))^(1/2)*(512*a^4*b^25*d^4*e^10 + 4608*a^6*b^23*d^4*e^10 + 17920*a^8*
b^21*d^4*e^10 + 38400*a^10*b^19*d^4*e^10 + 46080*a^12*b^17*d^4*e^10 + 21504*a^14*b^15*d^4*e^10 - 21504*a^16*b^
13*d^4*e^10 - 46080*a^18*b^11*d^4*e^10 - 38400*a^20*b^9*d^4*e^10 - 17920*a^22*b^7*d^4*e^10 - 4608*a^24*b^5*d^4
*e^10 - 512*a^26*b^3*d^4*e^10))/(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4
+ 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*d^4)) + ((e*cot(c + d*x))^(1/2)*(72*a*b^22
*d^2*e^9 + 576*a^3*b^20*d^2*e^9 + 5024*a^5*b^18*d^2*e^9 + 14272*a^7*b^16*d^2*e^9 + 27824*a^9*b^14*d^2*e^9 + 53
184*a^11*b^12*d^2*e^9 + 70240*a^13*b^10*d^2*e^9 + 47680*a^15*b^8*d^2*e^9 + 12616*a^17*b^6*d^2*e^9 - 64*a^21*b^
2*d^2*e^9))/(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 +
56*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*d^4)) - (90*a*b^19*d^2*e^9 + 846*a^3*b^17*d^2*e^9 + 1714*a^5*b
^15*d^2*e^9 + 3606*a^7*b^13*d^2*e^9 - 14578*a^9*b^11*d^2*e^9 - 34486*a^11*b^9*d^2*e^9 - 14970*a^13*b^7*d^2*e^9
+ 2258*a^15*b^5*d^2*e^9 - 32*a^17*b^3*d^2*e^9)/(a^20*d^5 + a^4*b^16*d^5 + 8*a^6*b^14*d^5 + 28*a^8*b^12*d^5 +
56*a^10*b^10*d^5 + 70*a^12*b^8*d^5 + 56*a^14*b^6*d^5 + 28*a^16*b^4*d^5 + 8*a^18*b^2*d^5)) + ((e*cot(c + d*x))^
(1/2)*(18*a^2*b^15*e^8 - 9*b^17*e^8 - 71*a^4*b^13*e^8 + 892*a^6*b^11*e^8 + 857*a^8*b^9*e^8 + 6802*a^10*b^7*e^8
- 1257*a^12*b^5*e^8))/(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^1
2*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^2*d^4))*1i - (1i/(4*(b^6*d^2*e - a^6*d^2*e - 15*a^2*b
^4*d^2*e - a^3*b^3*d^2*e*20i + 15*a^4*b^2*d^2*e + a*b^5*d^2*e*6i + a^5*b*d^2*e*6i)))^(1/2)*((1i/(4*(b^6*d^2*e
- a^6*d^2*e - 15*a^2*b^4*d^2*e - a^3*b^3*d^2*e*20i + 15*a^4*b^2*d^2*e + a*b^5*d^2*e*6i + a^5*b*d^2*e*6i)))^(1/
2)*((1i/(4*(b^6*d^2*e - a^6*d^2*e - 15*a^2*b^4*d^2*e - a^3*b^3*d^2*e*20i + 15*a^4*b^2*d^2*e + a*b^5*d^2*e*6i +
a^5*b*d^2*e*6i)))^(1/2)*((1i/(4*(b^6*d^2*e - a^6*d^2*e - 15*a^2*b^4*d^2*e - a^3*b^3*d^2*e*20i + 15*a^4*b^2*d^
2*e + a*b^5*d^2*e*6i + a^5*b*d^2*e*6i)))^(1/2)*((192*a^2*b^24*d^4*e^10 + 1728*a^4*b^22*d^4*e^10 + 8320*a^6*b^2
0*d^4*e^10 + 27264*a^8*b^18*d^4*e^10 + 62592*a^10*b^16*d^4*e^10 + 99456*a^12*b^14*d^4*e^10 + 107520*a^14*b^12*
d^4*e^10 + 76800*a^16*b^10*d^4*e^10 + 33984*a^18*b^8*d^4*e^10 + 7872*a^20*b^6*d^4*e^10 + 384*a^22*b^4*d^4*e^10
- 128*a^24*b^2*d^4*e^10)/(a^20*d^5 + a^4*b^16*d^5 + 8*a^6*b^14*d^5 + 28*a^8*b^12*d^5 + 56*a^10*b^10*d^5 + 70*
a^12*b^8*d^5 + 56*a^14*b^6*d^5 + 28*a^16*b^4*d^5 + 8*a^18*b^2*d^5) + ((1i/(4*(b^6*d^2*e - a^6*d^2*e - 15*a^2*b
^4*d^2*e - a^3*b^3*d^2*e*20i + 15*a^4*b^2*d^2*e + a*b^5*d^2*e*6i + a^5*b*d^2*e*6i)))^(1/2)*(e*cot(c + d*x))^(1
/2)*(512*a^4*b^25*d^4*e^10 + 4608*a^6*b^23*d^4*e^10 + 17920*a^8*b^21*d^4*e^10 + 38400*a^10*b^19*d^4*e^10 + 460
80*a^12*b^17*d^4*e^10 + 21504*a^14*b^15*d^4*e^10 - 21504*a^16*b^13*d^4*e^10 - 46080*a^18*b^11*d^4*e^10 - 38400
*a^20*b^9*d^4*e^10 - 17920*a^22*b^7*d^4*e^10 - 4608*a^24*b^5*d^4*e^10 - 512*a^26*b^3*d^4*e^10))/(a^20*d^4 + a^
4*b^16*d^4 + 8*a^6*b^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16
*b^4*d^4 + 8*a^18*b^2*d^4)) - ((e*cot(c + d*x))^(1/2)*(72*a*b^22*d^2*e^9 + 576*a^3*b^20*d^2*e^9 + 5024*a^5*b^1
8*d^2*e^9 + 14272*a^7*b^16*d^2*e^9 + 27824*a^9*b^14*d^2*e^9 + 53184*a^11*b^12*d^2*e^9 + 70240*a^13*b^10*d^2*e^
9 + 47680*a^15*b^8*d^2*e^9 + 12616*a^17*b^6*d^2*e^9 - 64*a^21*b^2*d^2*e^9))/(a^20*d^4 + a^4*b^16*d^4 + 8*a^6*b
^14*d^4 + 28*a^8*b^12*d^4 + 56*a^10*b^10*d^4 + 70*a^12*b^8*d^4 + 56*a^14*b^6*d^4 + 28*a^16*b^4*d^4 + 8*a^18*b^
2*d^4)) - (90*a*b^19*d^2*e^9 + 846*a^3*b^17*d^2*e^9 + 1714*a^5*b^15*d^2*e^9 + 3606*a^7*b^13*d^2*e^9 - 14578*a^
9*b^11*d^2*e^9 - 34486*a^11*b^9*d^2*e^9 - 14970*a^13*b^7*d^2*e^9 + 2258*a^15*b^5*d^2*e^9 - 32*a^17*b^3*d^2*e^9
)/(a^20*d^5 + a^4*b^16*d^5 + 8*a^6*b^14*d^5 + 2...

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